Abstract-We study the watersheds in edge-weighted graphs. We define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface. We first establish the consistency of these watersheds: they can be equivalently defined by their "catchment basins" (through a steepest descent property) or by the "dividing lines" separating these catchment basins (through the drop of water principle). Then we prove, through an equivalence theorem, their optimality in terms of minimum spanning forests. Afterward, we introduce a lineartime algorithm to compute them. To the best of our knowledge, similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm, both in theory and practice. Finally, the defined concepts are illustrated in image segmentation leading to the conclusion that the proposed approach improves, on the tested images, the quality of watershed-based segmentations.

Abstract-We recently introduced watershed cuts, a notion of watershed in edge-weighted graphs. In this paper, our main contribution is a thinning paradigm from which we derive three algorithmic watershed cut strategies: The first one is well suited to parallel implementations, the second one leads to a flexible linear-time sequential implementation, whereas the third one links the watershed cuts and the popular flooding algorithms. We state that watershed cuts preserve a notion of contrast, called connection value, on which several morphological region merging methods are (implicitly) based. We also establish the links and differences between watershed cuts, minimum spanning forests, shortest path forests, and topological watersheds. Finally, we present illustrations of the proposed framework to the segmentation of artwork surfaces and diffusion tensor images.

We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G. Afterward, we propose several new openings, closings, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of G and (ii) on the subgraphs of G. The proposed framework is then extended to functions that weight the vertices and edges of a graph. We illustrate with applications to binary and grayscale image denoising, for which, on the provided images, the proposed approach outperforms the usual one based on structuring elements.

Abstract. We study hierarchical segmentations that are optimal in the sense of minimal spanning forests of the original image. We introduce a region-merging operation called uprooting, and we prove that optimal hierarchical segmentations are equivalent to the ones given by uprooting a watershed-cut based segmentation. Based on those theoretical results, we propose an efficient algorithm to compute such hierarchies, as well as the first saliency map algorithm compatible with the morphological filtering framework.

Abstract. The goal of this paper is to provide linear or quasi-linear algorithms for producing some of the various trees used in mathemetical morphology, in particular the trees corresponding to hierarchies of watershed cuts and hierarchies of constrained connectivity. A specific binary tree, corresponding to an ordered version of the edges of the minimum spanning tree, is the key structure in this study, and is computed thanks to variations around Kruskal algorithm for minimum spanning tree.

Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and we characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasiflat zones preservation. In practice, these results make up a toolkit for designing new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies. More precisely, we show the practical interest of the proposed framework for: i) hierarchical watershed image segmentations, ii) combinations of different hierarchical segmentations, iii) hierarchicalizations of some non-hierarchical image segmentation methods based on regional dissimilarities, and iv) hierarchical analysis of geographical data.

Abstract. In edge-weighted graphs, we provide a unified presentation of a family of popular morphological hierarchies such as component trees, quasi flat zones, binary partition trees, and hierarchical watersheds. For any hierarchy of this family, we show if (and how) it can be obtained from any other element of the family. In this sense, the main contribution of this paper is the study of all constructive links between these hierarchies.

Abstract. In this article, we present the work towards improving the overall workflow of the Percutaneous Coronary Interventions (PCI) procedures by capacitating the imaging instruments to precisely monitor the steps of the procedure. In the long term, such capabilities can be used to optimize the image acquisition to reduce the amount of dose or contrast media employed during the procedure. We present the automatic VOIDD algorithm to detect the vessel of intervention which is going to be treated during the procedure by combining information from the vessel image with contrast agent injection and images acquired during guidewire tip navigation. Due to the robust guidewire tip segmentation method, this algorithm is also able to automatically detect the sequence corresponding to guidewire navigation. We present an evaluation methodology which characterizes the correctness of the guide wire tip detection and correct identification of the vessel navigated during the procedure. On a dataset of 2213 images from 8 sequences of 4 patients, VOIDD identifies vesselof-intervention with accuracy in the range of 88% or above and absence of tip with accuracy in range of 98% or above depending on the test case.